cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A059680 Triangle T(n,k) read by rows giving number of fixed 4 X k polyominoes with n cells (n >= 4, 1<=k<=n-3).

Original entry on oeis.org

1, 0, 12, 0, 18, 50, 0, 8, 154, 120, 0, 1, 212, 584, 230, 0, 0, 158, 1396, 1526, 388, 0, 0, 62, 2038, 5154, 3276, 602, 0, 0, 12, 1952, 11328, 14192, 6194, 880, 0, 0, 1, 1232, 17598, 41196, 32824, 10704, 1230, 0, 0, 0, 488, 19912, 87980, 117616, 67284, 17294, 1660
Offset: 4

Views

Author

N. J. A. Sloane, Feb 05 2001

Keywords

Examples

			Triangle starts:
1;
0, 12;
0, 18,  50;
0,  8, 154,  120;
0,  1, 212,  584,   230;
0,  0, 158, 1396,  1526,   388;
0,  0,  62, 2038,  5154,  3276,  602;
0,  0,  12, 1952, 11328, 14192, 6194, 880;
...

Links

Crossrefs

Column sums are A034187.
Cf. A059678, A059679, A059681, A059684.

Formula

T(n,k) = 0 for n > 4*k. - Andrew Howroyd, Oct 02 2017

Extensions

Terms a(32) and beyond from Andrew Howroyd, Oct 02 2017

A059678 Triangle T(n,k) giving number of fixed 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).

Original entry on oeis.org

1, 0, 4, 0, 1, 8, 0, 0, 6, 12, 0, 0, 1, 18, 16, 0, 0, 0, 8, 38, 20, 0, 0, 0, 1, 32, 66, 24, 0, 0, 0, 0, 10, 88, 102, 28, 0, 0, 0, 0, 1, 50, 192, 146, 32, 0, 0, 0, 0, 0, 12, 170, 360, 198, 36, 0, 0, 0, 0, 0, 1, 72, 450, 608, 258, 40, 0, 0, 0, 0, 0, 0, 14, 292, 1002, 952, 326, 44, 0, 0, 0
Offset: 2

Views

Author

N. J. A. Sloane, Feb 05 2001

Keywords

Examples

			Triangle begins:
1;
0, 4;
0, 1, 8;
0, 0, 6, 12;
0, 0, 1, 18, 16;
0, 0, 0,  8, 38, 20;
0, 0, 0,  1, 32, 66, 24;
...

Links

Crossrefs

Column sums are A034182.
Cf. A059679, A059680, A059681, A059682.

Programs

  • Maple
    with(combinat): for n from 2 to 30 do for k from 1 to n-1 do printf(`%d,`,sum(binomial(n-k+1, 2*k-n-v)*binomial(n-k+v, n-k), v=0..k) ) od:od:
  • Mathematica
    t[n_, k_] := Sum[Binomial[n-k+1, 2*k-n-v]*Binomial[n-k+v, n-k], {v, 0, k}]; Table[t[n, k], {n, 2, 15}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Dec 20 2013 *)

Formula

T(n, k) = Sum_v C(n-k+1, 2*k-n-v)*C(n-k+v, n-k).
G.f. (1+x*y)^2/(1-x*y)*1/((1-x*y)-(1+x*y)*x^2*y). - Christopher Hanusa (chanusa(AT)math.washington.edu), Sep 22 2004
T(n,k) = 0 for n > 2*k. - Andrew Howroyd, Oct 02 2017

Extensions

More terms from James Sellers, Feb 06 2001

A059679 Triangle T(n,k) giving number of fixed 3 X k polyominoes with n cells (n >= 3, 1<=k<=n-2).

Original entry on oeis.org

1, 0, 8, 0, 6, 25, 0, 1, 44, 50, 0, 0, 32, 154, 83, 0, 0, 9, 212, 376, 124, 0, 0, 1, 158, 784, 750, 173, 0, 0, 0, 62, 987, 2133, 1316, 230, 0, 0, 0, 12, 778, 3802, 4803, 2114, 295, 0, 0, 0, 1, 370, 4622, 11127, 9490, 3184, 368
Offset: 3

Views

Author

N. J. A. Sloane, Feb 05 2001

Keywords

Examples

			Triangle starts:
1;
0, 8;
0, 6, 25;
0, 1, 44,  50;
0, 0, 32, 154,  83;
0, 0,  9, 212, 376,  124;
0, 0,  1, 158, 784,  750,  173;
0, 0,  0,  62, 987, 2133, 1316, 230;
...

Links

Crossrefs

Column sums are A034184.
Cf. A059678, A059680, A059681, A059683.

Formula

T(n,k) = 0 for n > 3*k. - Andrew Howroyd, Oct 02 2017

Extensions

a(8) corrected and terms a(39) and beyond from Andrew Howroyd, Oct 02 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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